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Difference of Two Squares (Longer Expressions)Difference of Two Squares (Longer Expressions)
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Question 1 of 6
1. Question
Expand and simplify.`(x-2)^2-(x-2)(x+2)`Hint
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Expanding Perfect Square Binomials
$$(\color{#00880A}{a}-\color{#9a00c7}{b})^2=\color{#00880A}{a}^2-2\color{#00880A}{a}\color{#9a00c7}{b}+\color{#9a00c7}{b}^2$$Difference of Two Squares
$$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})=\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$$First, expand the values inside the first bracket by Expanding Perfect Square Binomials$$(\color{#00880A}{x}-\color{#9a00c7}{2})^2 \color{#9E9E9E}{-(x-2)(x+2)}$$ `=` $$\color{#00880A}{x}^2-2(\color{#00880A}{x})(\color{#9a00c7}{2})+\color{#9a00c7}{2}^2 \color{#9E9E9E}{-(x-2)(x+2)}$$ `=` `x^2-4x+4-(x-2)(x+2)` Simplify Next, expand the values inside the remaining brackets using the Difference of Two Squares$$\color{#9E9E9E}{x^2-4x+4\;-\;}(\color{#00880A}{x}-\color{#9a00c7}{2})(\color{#00880A}{x}+\color{#9a00c7}{2})$$ `=` $$\color{#9E9E9E}{x^2-4x+4\;-\;(}\color{#00880A}{x}^2-\color{#9a00c7}{2}^2\color{#9E9E9E}{)}$$ `=` `x^2-4x+4-(x^2-4)` Simplify `=` `x^2-4x+4-x^2+4` Finally, simplify further by combining like terms.`x^2` `-4x+4` `-x^2` `+4` `=` `-4x` `+4+4` `x^2-x^2` cancels out `=` `-4x+8` `-4x+8` -
Question 2 of 6
2. Question
Expand and simplify.`28x-(x-2)(x-5)+x^2`Hint
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First, use the FOIL Method to expand and simplify the values inside the brackets.Multiply the First, Outside, Inside, and Last terms, then simplify.$$\color{#9E9E9E}{28x-[}\color{#00880A}{(x)(x)}+\color{#9a00c7}{(x)(-5)}+\color{#007DDC}{(-2)(x)}+\color{#e65021}{(-2)(-5)}\color{#9E9E9E}{]+x^2}$$ `=` $$\color{#9E9E9E}{28x-(}x^2-5x-2x+10\color{#9E9E9E}{)+x^2}$$ `=` `28x-(x^2-7x+10)+x^2` Collect like terms `=` `28x-x^2+7x-10+x^2` Finally, simplify further by combining like terms.`28x` `-x^2` `+7x-10` `+x^2` `=` `28x+7x` `-10` `x^2-x^2` cancels out `=` `35x-10` `35x-10` -
Question 3 of 6
3. Question
Expand and simplify.`(b-a)(b-t)-b^2+bt`Hint
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First, use the FOIL Method to expand and simplify the values inside the brackets.Multiply the First, Outside, Inside, and Last terms, then simplify.$$\color{#00880A}{(b)(b)}+\color{#9a00c7}{(b)(-t)}+\color{#007DDC}{(-a)(b)}+\color{#e65021}{(-a)(-t)}\color{#9E9E9E}{-b^2+bt}$$ `=` `b^2-bt-ab+at-b^2+bt` Finally, simplify further by combining like terms.`b^2` `-bt-ab+at``-b^2``+bt` `=` `-bt` `-ab+at` `+bt` `b^2-b^2` cancels out `=` `-ab+at` `-bt+bt` cancels out `=` `at-ab` Rearrange the values `at-ab` -
Question 4 of 6
4. Question
Expand and simplify.`(2x+3)(6x-2)-5(x-4)(x+4)`Hint
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Fantastic!
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Difference of Two Squares
$$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})=\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$$First, use the FOIL Method to expand and simplify the values inside the first two brackets.Multiply the First, Outside, Inside, and Last terms, then simplify.$$\color{#00880A}{(2x)(6x)}+\color{#9a00c7}{(2x)(-2)}+\color{#007DDC}{(3)(6x)}+\color{#e65021}{(3)(-2)}\color{#9E9E9E}{-5(x-4)(x+4)}$$ `=` $$12x^2\color{#CC0000}{-4x+18x}-6\color{#9E9E9E}{-5(x-4)(x+4)}$$ Collect like terms `=` `12x^2+14x-6-5(x-4)(x+4)` Next, substitute the values inside the remaining brackets into the formula given for Difference of Two Squares.$$\color{#9E9E9E}{12x^2+14x-6-5}(\color{#00880A}{x}-\color{#9a00c7}{4})(\color{#00880A}{x}+\color{#9a00c7}{4})$$ `=` $$\color{#9E9E9E}{12x^2+14x-6-5(}\color{#00880A}{x}^2-\color{#9a00c7}{4}^2\color{#9E9E9E}{)}$$ `=` `12x^2+14x-6-5(x^2-16)` Then, distribute `5` to each value inside the bracket.`12^2+14x-6-` `5``(x^2-16)` `=` `12^2+14x-6-(``5``timesx^2)-(``5``times16)` `=` `12^2+14x-6-(5x^2-80)` `=` `12^2+14x-6-5x^2+80` Finally, simplify further by combining like terms.`12x^2` `+14x-6` `-5x^2` `+80` `=` `7x^2+14x` `-6+80` `=` `7x^2+14x+74` `7x^2+14x+74` -
Question 5 of 6
5. Question
Expand and simplify.`-10m-(m-5)^2`Hint
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Keep Going!
Incorrect
Expanding Perfect Square Binomials
$$(\color{#00880A}{a}-\color{#9a00c7}{b})^2=\color{#00880A}{a}^2-2\color{#00880A}{a}\color{#9a00c7}{b}+\color{#9a00c7}{b}^2$$Difference of Two Squares
$$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})=\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$$First, expand the values inside the bracket by Expanding Perfect Square Binomials$$\color{#9E9E9E}{-\;10m\;-\;}(\color{#00880A}{m}-\color{#9a00c7}{5})^2$$ `=` $$\color{#9E9E9E}{-\;10m\;-\;}\color{#00880A}{m}^2-2(\color{#00880A}{m})(\color{#9a00c7}{5})+\color{#9a00c7}{5}^2$$ `=` `-10m-(m^2-10m+25)` Simplify `=` `-10m-m^2+10m-25` Finally, simplify further by combining like terms.`-10m` `-m^2` `+10m` `-25` `=` `-m^2-25` `-10m+10m` cancels out `-m^2-25` -
Question 6 of 6
6. Question
Expand and simplify.`(y-2)^2+7y(y-2)(y+2)`Hint
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Great Work!
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Expanding Perfect Square Binomials
$$(\color{#00880A}{a}-\color{#9a00c7}{b})^2=\color{#00880A}{a}^2-2\color{#00880A}{a}\color{#9a00c7}{b}+\color{#9a00c7}{b}^2$$Difference of Two Squares
$$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})=\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$$First, expand the values inside the first bracket by Expanding Perfect Square Binomials$$(\color{#00880A}{y}-\color{#9a00c7}{2})^2 \color{#9E9E9E}{+7y(y-2)(y+2)}$$ `=` $$\color{#00880A}{y}^2-2(\color{#00880A}{y})(\color{#9a00c7}{2})+\color{#9a00c7}{2}^2 \color{#9E9E9E}{+7y(y-2)(y+2)}$$ `=` `y^2-4y+4+7y(y-2)(y+2)` Simplify Next, expand the values inside the remaining brackets using the Difference of Two Squares$$\color{#9E9E9E}{y^2-4y+4+7y}(\color{#00880A}{y}-\color{#9a00c7}{2})(\color{#00880A}{y}+\color{#9a00c7}{2})$$ `=` $$\color{#9E9E9E}{y^2-4y+4+7y(}\color{#00880A}{y}^2-\color{#9a00c7}{2}^2\color{#9E9E9E}{)}$$ `=` `y^2-4y+4+7y(y^2-4)` Simplify Then, distribute `7y` to each value inside the bracket.`y^2-4y+4+``7y``(y^2-4)` `=` `y^2-4y+4+(``7y``timesy^2)-(``7y``times4)` `=` `y^2-4y+4+7y^3-28y` Finally, simplify further by combining like terms.`y^2` `-4y` `+4+7y^3` `-28y` `=` `y^2-32y+4+7y^3` `=` `7y^3+y^2-32y+4` Rearrange the values `7y^3+y^2-32y+4`